Comparison of Some Purities, Flatnesses and Injectivities

Abstract

In this article, we compare (n, m)-purities for different pairs of positive integers (n, m). When R is a commutative ring, these purities are not equivalent if R does not satisfy the following property: there exists a positive integer p such that, for each maximal ideal P, every finitely generated ideal of R _P is p-generated. When this property holds, then the (n, m)-purity and the (n, m′)-purity are equivalent if m and m′ are integers ≥np. These results are obtained by a generalization of Warfield's methods. There are also some interesting results when R is a semiperfect strongly π-regular ring. We also compare (n, m)-flatnesses and (n, m)-injectivities for different pairs of positive integers (n, m). In particular, if R is right perfect and right self (ℵ₀, 1)-injective, then each (1, 1)-flat right R-module is projective. In several cases, for each positive integer p, all (n, p)-flatnesses are equivalent. But there are some examples where the (1, p)-flatness is not equivalent to the (1, p + 1)-flatness.

Key Words:

• (n
• m)-Coherent ring
• (n
• m)-Flat module
• (n
• m)-Injective module
• (n
• m)-Pure submodule

2000 Mathematics Subject Classification:

• Primary 16D40
• 16D50
• 16D80

Notes

Communicated by T. Albu.